Classifications

• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L25/00—Baseband systems
  • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
  • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
  • H04L25/03006—Arrangements for removing intersymbol interference
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04B—TRANSMISSION
  • H04B15/00—Suppression or limitation of noise or interference
• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
  • G06F17/10—Complex mathematical operations
• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M1/00—Analogue/digital conversion; Digital/analogue conversion
  • H03M1/06—Continuously compensating for, or preventing, undesired influence of physical parameters
  • H03M1/0617—Continuously compensating for, or preventing, undesired influence of physical parameters characterised by the use of methods or means not specific to a particular type of detrimental influence
  • H03M1/0626—Continuously compensating for, or preventing, undesired influence of physical parameters characterised by the use of methods or means not specific to a particular type of detrimental influence by filtering
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04B—TRANSMISSION
  • H04B1/00—Details of transmission systems, not covered by a single one of groups H04B3/00 - H04B13/00; Details of transmission systems not characterised by the medium used for transmission
  • H04B1/69—Spread spectrum techniques
  • H04B1/7163—Spread spectrum techniques using impulse radio
  • H04B1/71637—Receiver aspects
• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M1/00—Analogue/digital conversion; Digital/analogue conversion
  • H03M1/12—Analogue/digital converters
  • H03M1/124—Sampling or signal conditioning arrangements specially adapted for A/D converters
  • H03M1/1245—Details of sampling arrangements or methods
  • H03M1/1265—Non-uniform sampling
  • H03M1/127—Non-uniform sampling at intervals varying with the rate of change of the input signal
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/26—Systems using multi-frequency codes
  • H04L27/2601—Multicarrier modulation systems
  • H04L27/2647—Arrangements specific to the receiver only

Definitions

• This application relates generally to signal processing and more specifically, but not exclusively, to wireless telecommunication, signal acquisition and reconstruction.
• the disclosure relates in some aspects to acquiring signals, including signals from outside of the subspace of bandlimited signals associated with the Shannon theorem, while still providing an acceptable reconstruction. In some aspects this may involve taking advantage of some sort of sparsity in the signal being reconstructed.
• sparse sampling at a rate characterized by how sparse signal components are per unit of time, the Nyquist constraints may be avoided while accurately sampling and reconstructing signals.
• sampling is performed at the rate of innovation of the signal.
• the disclosure relates in some aspects to using a denoising process in conjunction with sparse sampling techniques.
• a denoising process utilizing a Cadzow algorithm may be used to reduce the amount of noise associated with sampled information.
• the denoising process may be iterative such that the denoising process is repeated until the samples are denoised to a sufficient degree. This denoising process is especially useful when applied to signal sampled at sub- Nyquist rates, but can also be used with bandlimited signals sampled at rates higher than Nyquist.
• the denoising process converts a set of received samples into another set corresponding to a signal with a Finite Rate of Innovation (FRI), or to an approximation of such a signal.
• the denoising process thus removes or reduces at least one component of the noise, i.e., the component that makes the signal have a non Finite Rate of Innovation.
• the parameters (amplitude (weight) and location (phase, shift)) of the set of samples delivered by the denoising process may still be noisy, but the signal-to-noise ratio is improved by this process.
• the output of an infinite number of iterations of Cadzow is thus not only “denoised” but is actually a signal with a Finite Rate of Innovation, very unlike the noisy signal.
• the output may be an FRI signal (e.g., on the order of le-10).
• Such an FRI signal may equivalently be represented, either by a sequence of uniform samples, or by the set of parameters (innovations) that provide these samples.
• the switch between the two representations is performed by using an annihilation filter technique. In the disclosed approach this technique may be reduced to be the inverse operation of the sampling operation.
• a subspace technique can be used for the retrieval of a parametric model.
• the matrix of coefficients derived from the samples has a structure and rank condition which can be taken advantage of.
• the matrix of Fourier coefficients is Toeplitz and has rank K when there are K Diracs in the time domain signal.
• a singular value decomposition can be used to get a rank K approximation to the noisy matrix, which is a subspace approximation.
• the disclosure relates in some aspects to a method where a signal with a non Finite Rate of Innovation, for example a noisy signal, can be acquired through sampling and projected onto the subspace of signals with a Finite Rate of Innovation, allowing a perfect or at least improved interpolation from the samples.
• a signal with a non Finite Rate of Innovation for example a noisy signal
• the disclosure relates in some aspects to determining the number of samples to obtain from a signal over a period of time based on noise associated with the signal. For example, the number of samples may be selected based on the known or assumed signal-to-noise ratio, and/or on the desired accuracy of the reconstructed signal.
• the disclosure relates in some aspects to determining (e.g., defining) a sampling kernel to be used to sample the signal based on noise associated with the signal. For example, the amount and type of noise in a signal may affect the bandwidth of a sampling kernel and/or the type of a sampling kernel to be used to sample the signal. The bandwidth of the sampling kernel, in turn, may affect the number of samples to acquire.
• the disclosure relates in some aspects to determining the finite number of innovations of a received signal. For example, a number of innovations of a signal may be determined based on at least one rank of at least one matrix that is defined based on the received signal. In some aspects such a matrix may be associated with an annihilation filter.
• the disclosure also relates in some aspects to combination of a denoising process with annihilating filter methods in order to retrieve information from a noisy, sparse sampled signal.
• FIG. 1 is a simplified block diagram of several sample aspects of a communication receiver, with schematic indications of potential noise perturbations in the analog part and in the digital part.
• FIG. 2 is a simplified block diagram of several sample aspects of a communication receiver, with schematic indications of potential noise perturbations in the analog part and in the digital part;
• FIG. 3 is a simplified block diagram of several sample aspects of a communication system
• FIG. 4 is a simplified block diagram of several sample aspects of communication components
• FIGS. 5A and 5B are simplified block diagrams of several sample aspects of apparatuses configured to provide signal acquisition and reconstruction as taught herein.
• the various features illustrated in the drawings may not be drawn to scale. Accordingly, the dimensions of the various features may be arbitrarily expanded or reduced for clarity. In addition, some of the drawings may be simplified for clarity. Thus, the drawings may not depict all of the components of a given apparatus (e.g., device) or method.
• like reference numerals may be used to denote like features throughout the specification and figures.
• FIG. 1 relates to a system that may be employed for a noiseless case and, potentially, for a substantially noiseless case (e.g., where the effect of noise may be ignored or will be automatically compensated).
• the system comprises a transmission channel which possibly adds an analog noise ⁇ (t) to a transmitted signal (the "another" signal) x(t).
• the analog signal y(t) transmitted over the transmission channel is received by a suitable receiver 305 in an apparatus 302.
• the receiver 305 may comprise for example an antenna and a radio frequency part (not shown).
• the apparatus 302 may comprise a sampler that samples a received signal y(t) using a sampling kernel.
• a sampling kernel for example a sine filter, with a suitable bandwidth.
• the filtered signal is then sampled at a sample frequency 1/T, and possibly quantized.
• the sampler may comprise at least a portion of the sampling filter.
• the sampling kernel e.g., a filter based on the sampling kernel
• the sampler may be implemented in a circuit in the apparatus 302.
• the quantization process, error corrections or various perturbations in the receiver 305 or elsewhere in the apparatus 302 may add a digital noise ⁇ (n) to the sampled signal y(n) (which may be referred to herein as y n ).
• the "addition block" is simply intended to illustrate, for example, that at some point digital noise ⁇ (n) may be imparted on the sampled signal y(n) whereby the input to an estimator 303 may be a composite signal comprising y(n) and ⁇ (n).
• the analog noise ⁇ (t) and the digital noise ⁇ (n) are both null or at least low compared to the signal, so that a denoising process may not be needed, as will be described.
• the sampled signal y(n) with the additional digital noise ⁇ (n) is then converted in the frequency domain, using for example a Fast Fourier Transformation (FFT) or another suitable transformation.
• FFT Fast Fourier Transformation
• An annihilating filter, as part of the estimator 303, is then computed in order to retrieve shifts t k , and then weights X k , by resolving a linear system in the estimator.
• the original signal x(t) may then be accurately reconstructed from the set of retrieved innovations t ⁇ , X k (e.g., from estimates of the shifts and weights).
• a complete reconstruction of the another signal x(t) (which may be referred to herein as x t ) is not required and it may be sufficient to retrieve some information ("innovations") corresponding to this signal, for example only the shifts t ⁇ , only the weights X k , and/or other information relating to x(t).
• FIG. 2 relates to a system that may be employed for the case of noisy signals.
• the estimator 303 also comprises a denoiser for replacing the sampled signal y(n) (or the FFT transform of this signal) by an approximation with no or less noise, i.e. a "denoised" sequence of samples (y' n ).
• denoised means that at least a part of the noise has been removed or reduced.
• a denoiser is thus an apparatus, part or software module that converts a noisy signal into another signal from which a better, less noisy estimation of the required information can be computed.
• the annihilating filter and the linear system are shown as being implemented in a processor component 306.
• the weights X k and shifts t k , or other information relating to the another signal x(t), are thus retrieved from the denoised signal y'(n) (which may be referred to herein as y' n ) using a similar annihilating filter and by solving a similar linear system.
• the denoising may be applied to the samples in the time domain or, as illustrated, in the frequency domain after a suitable transformation such as a Fast Fourier Transformation (FFT).
• FFT Fast Fourier Transformation
• the sampling rate 1/T used for sampling the (filtered) received signal y(t) (which may be referred to herein as y t ) may be sub- Nyquist, i.e. lower than the minimal sampling rate given by the Shannon theorem applied to the another signal x(t) and/or to the received signal y(t), while still allowing an accurate or (if the signal-to-noise ratio is sufficient) even perfect reconstruction of the another signal x(t).
• One of the options associated with the sampling rate is to choose it higher than the finite rate of innovation p of the another signal x(t).
• the method thus even applies to a non bandlimited signal, such as a sequence of Diracs, a rectangular signal, a piecewise polynomial signal, any signal with a finite duration and finite number of innovations during this duration, any periodic signal with a finite number of innovations during each period, etc., or to a signal y(t) that includes a noise component making it not bandlimited.
• a non bandlimited signal such as a sequence of Diracs, a rectangular signal, a piecewise polynomial signal, any signal with a finite duration and finite number of innovations during this duration, any periodic signal with a finite number of innovations during each period, etc., or to a signal y(t) that includes a noise component making it not bandlimited.
• the sampling rate 1/T used for sampling the received signal y(t) thus depends mainly on the rate of innovation of the another signal x(t). This rate of innovation may be previously known, estimated, assumed, or retrieved from the signal itself, for example in a receiver.
• the sampling rate also depends on the level of noise ⁇ (t) and ⁇ (n), for example on the signal-to-noise ratio, which may also be previously known, estimated, assumed, or retrieved from the received signal.
• the determination of the sampling rate depends on the desired level of accuracy of the information one wants to retrieve.
• the determination of the sampling rate is based on a comparison between the signal-to-noise ratio of the reconstructed signal with an assumption of an expected signal to noise ratio.
• the sampling rate may be defined before the receipt of the another signal x(t) by the apparatus 302.
• the apparatus 302 may determine the sampling rate based on an estimate of the noise (e.g., estimated signal to noise ratio), a desired level of accuracy, a comparison of signal to noise ratios, or the rate of innovation (e.g., based on K which may be determined as discussed below).
• estimate of the noise e.g., estimated signal to noise ratio
• desired level of accuracy e.g., a desired level of accuracy
• comparison of signal to noise ratios e.g., based on K which may be determined as discussed below.
• sampling kernel ⁇ (t) in particular the bandwidth and/or function used as a sampling kernel, may also depend on the rate of innovation of the another signal x(t) and/or on noise, or on the signal-to-noise ratio, or on the type of noise, or more generally on an estimate of the analog noise ⁇ (t) and/or digital noise ⁇ (n).
• the apparatus 302 may define the kernel (e.g., by adjusting the bandwidth of an adjustable kernel filter) based on the noise (e.g., an estimate of noise) or the rate of innovation (e.g., based on K which may be determined as discussed below).
• the kernel e.g., by adjusting the bandwidth of an adjustable kernel filter
• the noise e.g., an estimate of noise
• the rate of innovation e.g., based on K which may be determined as discussed below.
• FIG. 3 illustrates a communication system with a transmitter 301 that sends an analog signal x(t) over a noisy communication channel, for example a wireless or wired communication channel.
• An apparatus 302 receives a signal y(t) based on x(t) and on noise ⁇ (t) added by the communication channel (e.g., the signal y(t) is a composite signal comprising x(t) and ⁇ (t)), and retrieves information corresponding to x(t) by sampling y(t) at a sub-Nyquist rate, possibly denoising the sampled signal, and applying suitable annihilating filter and linear system estimation methods as described below.
• a generalization of the space of bandlimited signals is the space of shift- invariant signals.
• the space of functions obtained by replacing sine with ⁇ in (1) defines a shift-invariant space S.
• p lim ⁇ ⁇ Cx /T, where Cx is the number of Diracs in the interval [-T/2, T/2].
• sampling theorem for the type of sparsely sampled signals is described below where p samples per unit of time are acquired, which perfectly describe the original signal, thereby allowing a perfect reconstruction of the original signal, as with the Shannon sampling procedure but with less samples or a lower sampling rate.
• the sampling rate should be (at least) p, the rate of innovation of the original signal. To show that that is sufficient can be done in a number of cases of interest.
• the archetypal sparse signal is the sum of Diracs, observed through a suitable sampling kernel. In this case, sampling theorems at the rate of innovation can be proven. Beyond the question of a representation theorem, efficient computational procedures, showing the practicality of the approach are described.
• a ⁇ -periodic stream of K Diracs ⁇ (t) with amplitudes Xk located at times tk e [0, ⁇ ] may be represented as:
• the weight X k and shifts t k each take K values.
• the signal x(t) is completely determined by the knowledge of the K amplitudes (weights) Xk and the K locations (shifts) tk, or equivalently, by the knowledge of Uk.
• the filter h m may thus be referred to as annihilating filter since it annihilates the discrete signal x m .
• the zeros of this filter uniquely define the locations tk of the
• the locations tk are retrieved from the zeros Uk of the z- transform in (7).
• the weights Xk are then obtained by considering, for instance, K consecutive Fourier-series coefficients as given in (6).
• a Vandermonde system of equations is provided which yields a unique solution for the weights Xk since the Uk's are distinct.
• no more than 2K consecutive coefficients x m may be needed to solve both the Toeplitz system (9) and the Vandermonde system. This confirms that the knowledge of only 2K
• Equation 12 [h 0 , h ls . . . , h L ] T the vector containing the coefficients of the annihilating filter, then Equation 12 is equivalent to:
• Equation 9 which can be seen as a rectangular extension of Equation 9.
• the rank of the matrix A does never exceed K.
• Equation 12 satisfied by the DFT coefficients y m is narrowly linked to the periodized sinc-Dirichlet window used prior to sampling.
• This approach may be generalized to other kernels such as for example the (non- periodized) sine, the Gaussian windows, and any window that satisfies a Strang-Fix like condition, i.e., the reproduction of polynomials.
• an increase in the sampling rate is performed to achieve robustness against noise.
• Equation 3 Equation 3
• Equation 4 Equation 4
• T ⁇ /N and ⁇ (t) is the Dirichlet kernel (Equation 5). Given that the rate of innovation of the signal is p, consider N > p ⁇ samples to fight the perturbation ⁇ n , making the data redundant by a factor of N/(p ⁇ ). Algorithms that may be applied to efficiently exploit this extra redundancy will now be discussed.
• One approach utilizes the Total Least-Squares approximation (implemented using a Singular Value Decomposition), optionally enhanced by an initial "denoising” (or “model matching") step provided by what is referred to herein as Cadzow 's iterated algorithm.
• Cadzow initial “denoising” (or “model matching") step provided by what is referred to herein as Cadzow 's iterated algorithm.
• the full algorithm, depicted in FIG. 2 is also detailed below in its two main components.
• the denoising process "projects" the noisy samples onto the sampled FRI model of Equation 15. It thus replaces the noisy digital samples by an approximation of the noiseless signal, or by a signal which may still be noisy (less noisy), but which has a finite rate of innovation, or is closer to a signal with a finite rate of innovation.
• the samples y n are very noisy, it is preferable to first denoise them by performing at least one iteration of Cadzow's algorithm before applying the above described total least square method or another method for retrieving the parameters (t ⁇ , X k ) of the other signal.
• the Cadzow process delivers a FRI signal, or substantially FRI signal, that can equivalently be represented, either by a sequence of uniform samples, or by a set of parameters (innovations) that provide these samples.
• the switch between the two representations is performed by using the Annihilating Filter technique, which may be reduced to the "inverse" operation of the sampling operation.
• the noiseless matrix A in Equation 13 is of rank K whenever L > K.
• step 2 Iterate step 2 until a condition is met, for example until the (K+ 1)* largest diagonal element of S is smaller than the K ⁇ largest diagonal element by a prerequisite factor depending on the requested accuracy and/or computing time.
• the iterations may also be repeated a predefined number of times, or until a desired signal to noise ratio of a reconstructed signal has been reached, or when a desired level of accuracy of the information relative to the another signal x(t) has been reached.
• the above described Cadzow denoising process outputs a signal with an improved signal-to-noise ratio, but in which the parameters (for example the location of the Diracs and their amplitude) are still noisy.
• the main advantage of this process is to deliver a signal which is parametric, i.e., a signal with a Finite Rate of Innovation, or at least a signal with a substantially Finite Rate of Innovation, or sufficiently close for the intended purpose to a signal with a Finite Rate of Innovation.
• the Cadzow process is thus not only a denoising process that merely improves the SNR ratio, in the way a Wiener filter for example would perform, but also projects the signal into the subspace of FRI signals for which reconstruction methods are available.
• denoising process and methods may be used in the framework of the disclosure.
• the singular value decomposition that can be used to get a rank K approximation to the noisy matrix, which is a subspace approximation.
• Other subspace techniques in particular variants based on the ESPRIT algorithm and/or on wavelets arrays, may be used for reducing the level of noise and for finding parameters of the signal one wants to reconstruct or for which one wants to retrieve information.
• the denoising and estimation of information relating to the another signal (x t ) may be combined in one common process.
• FIG. 4 depicts several sample components that may be employed to facilitate communication between devices.
• a first device 702 and a second device 704 are adapted to communicate via a communication link 706, for example a wireless communication link, over a suitable medium.
• a transmit (“TX”) data processor 708 receives traffic data (e.g., data packets) from a data buffer 710 or some other suitable component.
• the transmit data processor 708 processes (e.g., encodes, interleaves, and symbol maps) each data packet based on a selected coding and modulation scheme, and provides data symbols.
• a data symbol is a modulation symbol for data
• a pilot symbol is a modulation symbol for a pilot (which is known a priori).
• a modulator 712 receives the data symbols, pilot symbols, and possibly signaling for the reverse link, and performs modulation (e.g., OFDM, PPM or some other suitable modulation) and/or other processing as specified by the system, and provides a stream of output chips.
• a transmitter (“TMTR") 714 processes (e.g., converts to analog, filters, amplifies, and frequency upconverts) the output chip stream and generates a modulated signal, which is then transmitted from an antenna 716.
• the modulated signals transmitted by the device 702 (along with signals from other devices in communication with the device 704) are received by an antenna 718 of the device 704. Noise may be added during the transmission over the channel and/or during reception.
• a receiver (“RCVR”) 720 processes (e.g., conditions and digitizes) the received signal from the antenna 718 and provides received samples.
• a demodulator (“DEMOD”) 722 processes (e.g., demodulates and detects) the received samples and provides detected data symbols, which may be a noisy estimate of the data symbols transmitted to the device 704 by the other device(s).
• a receive (“RX”) data processor 724 processes (e.g., symbol demaps, deinterleaves, and decodes) the detected data symbols and provides decoded data associated with each transmitting device (e.g., device 702).
• a forward link Components involved in sending information from the device 704 to the device 702 (e.g., a forward link) will be now be treated.
• traffic data is processed by a transmit (“TX") data processor 726 to generate data symbols.
• a modulator 728 receives the data symbols, pilot symbols, and signaling for the forward link, performs modulation (e.g., OFDM or some other suitable modulation) and/or other pertinent processing, and provides an output chip stream, which is further conditioned by a transmitter (“TMTR") 730 and transmitted from the antenna 718.
• modulation e.g., OFDM or some other suitable modulation
• TMTR transmitter
• signaling for the forward link may include power control commands and other information (e.g., relating to a communication channel) generated by a controller 732 for all devices (e.g. terminals) transmitting on the reverse link to the device 704.
• the modulated signal transmitted by the device 704 is received together with noise by the antenna 716, conditioned and digitized by a receiver (“RCVR”) 734, and processed by a demodulator (“DEMOD”) 736 to obtain detected data symbols.
• a receive (“RX”) data processor 738 processes the detected data symbols and provides decoded data for the device 702 and the forward link signaling.
• a controller 740 receives power control commands and other information to control data transmission and to control transmit power on the reverse link to the device 704.
• the controllers 740 and 732 direct various operations of the device 702 and the device 704, respectively. For example, a controller may determine an appropriate filter, reporting information about the filter, and decode information using a filter.
• Data memories 742 and 744 may store program codes and data used by the controllers 740 and 732, respectively.
• FIG. 4 also illustrates that the communication components may include one or more components that perform operations as taught herein.
• a receive control component 746 may cooperate with the controller 740 and/or other components of the device 702 to receive information from another device (e.g., device 704).
• a receive control component 748 may cooperate with the controller 732 and/or other components of the device 704 to receive information from another device (e.g., device 702).
• a wireless device may include various components that perform functions based on signals that are transmitted by or received at the wireless device or other signals as taught herein.
• a wireless headset may include a transducer arranged to provide an audio output based on estimated information, retrieved parameters, or at least one sample as discussed above.
• a wireless watch may include a user interface arranged to provide an indication based estimated information, retrieved parameters, or at least one sample as discussed above.
• a wireless sensing device may include a sensor arranged to sense (e.g., to provide data to be transmitted) based on estimated information, retrieved parameters, or at least one sample as discussed above (e.g., received information that controls the sensing).
• the teachings herein may also be applied to optical or galvanic transmission channels, including optical transmission over optical fibers.
• a wireless device may communicate via one or more wireless communication links that are based on or otherwise support any suitable wireless communication technology.
• a wireless device may associate with a network.
• the network may comprise a body area network or a personal area network (e.g., an ultra-wideband network).
• the network may comprise a local area network or a wide area network.
• a wireless device may support or otherwise use one or more of a variety of wireless communication technologies, protocols, or standards such as, for example, UWB, CDMA, TDMA, OFDM, OFDMA, WiMAX, and Wi-Fi.
• a wireless device may support or otherwise use one or more of a variety of corresponding modulation or multiplexing schemes.
• a wireless device may thus include appropriate components (e.g., air interfaces) to establish and communicate via one or more wireless communication links using the above or other wireless communication technologies.
• a device may comprise a wireless transceiver with associated transmitter and receiver components that may include various components (e.g., signal generators and signal processors) that facilitate communication over a wireless medium.
• a wireless device may communicate via an impulse-based wireless communication link.
• an impulse-based wireless communication link may utilize ultra-wideband pulses that have a relatively short length (e.g., on the order of a few nanoseconds or less) and a relatively wide bandwidth.
• the ultra- wideband pulses may have a fractional bandwidth on the order of approximately 20% or more and/or have a bandwidth on the order of approximately 500 MHz or more.
• teachings herein may be incorporated into (e.g., implemented within or performed by) a variety of apparatuses (e.g., devices).
• a phone e.g., a cellular phone
• PDA personal data assistant
• an entertainment device e.g., a music or video device
• a headset e.g., headphones, an earpiece, etc.
• a microphone e.g., a medical sensing device (e.g., a biometric sensor, a heart rate monitor, a pedometer, an EKG device, a smart bandage, etc.)
• a user I/O device e.g., a watch, a remote control, a light switch, a keyboard, a mouse, etc.
• an environment sensing device e.g., a tire pressure monitor
• a computer e.g., a point-of-sale device, an entertainment device, a hearing aid, a set-top box, a "smar
• UWB ultra- wide band
• PPM pulse position modulation
• ADC analog-to-digital conversion
• a simple model of an UWB pulse is a Dirac convolved with a wideband, zero mean pulse. At the receiver, the signal is the convolution of the original pulse with the channel impulse response, which includes many reflections, and all this buried in high levels of noise.
• the methods and devices disclosed in this document are especially useful for sparse signals, i.e. signals with a low rate of innovation as compared to the bandwidth, be it in time and/or in space.
• the methods are however not limited to sparse signals and may be used for any signal with a finite rate of innovation, including bandlimited signal.
• the methods are not limited to discrete values; indeed, the innovation times t ⁇ and the weights C k used and retrieved by the methods may assume arbitrary real values.
• teachings herein may be adapted for use in low power applications (e.g., through the use of an impulse-based signaling scheme and low duty cycle modes) and may support a variety of data rates including relatively high data rates (e.g., through the use of high-bandwidth pulses).
• a wireless device may comprise an access device (e.g., a Wi- Fi access point) for a communication system.
• an access device may provide, for example, connectivity to another network (e.g., a wide area network such as the Internet or a cellular network) via a wired or wireless communication link.
• the access device may enable another device (e.g., a Wi-Fi station) to access the other network or some other functionality.
• another device e.g., a Wi-Fi station
• one or both of the devices may be portable or, in some cases, relatively non-portable.
• apparatuses 800, 900, 1000, and 1100 are represented as a series of interrelated functional blocks that may represent functions implemented by, for example, one or more integrated circuits (e.g., an ASIC) or may be implemented in some other manner as taught herein.
• an integrated circuit may include a processor, software, other components, or some combination thereof.
• the apparatuses 800, 900, 1000, and 1100 may include one or more modules that may perform one or more of the functions described above with regard to various figures.
• an ASIC for obtaining a digital signal 802, 902, or 1002 may correspond to, for example, a sampler and/or one or more other components as discussed herein.
• An ASIC for estimating information 804, 1008, or 1108 may correspond to, for example, a processor and/or one or more other components as discussed herein.
• An ASIC for determining a sampling rate 806 or 1118 may correspond to, for example, an estimator and/or one or more other components as discussed herein.
• An ASIC for defining a sampling kernel 808 or 1104 may correspond to, for example, an estimator and/or one or more other components as discussed herein.
• An ASIC for converting the digital signal 904 may correspond to, for example, a denoiser and/or one or more other components as discussed herein.
• An ASIC for retrieving parameters 906 may correspond to, for example, a processor and/or one or more other components as discussed herein.
• An ASIC for building a Toeplitz matrix 1004 may correspond to, for example, a processor and/or one or more other components as discussed herein.
• An ASIC for performing an SVD of the Toeplitz matrix 1006 may correspond to, for example, a processor and/or one or more other components as discussed herein.
• An ASIC for obtaining an analog signal 1102 may correspond to, for example, a receiver and/or one or more other components as discussed herein.
• An ASIC for using a sampling kernel 1106 may correspond to, for example, a sampler and/or one or more other components as discussed herein.
• An ASIC for defining an annihilating filter 1110 may correspond to, for example, a processor and/or one or more other components as discussed herein.
• An ASIC for reconstructing 1112 may correspond to, for example, a processor and/or one or more other components as discussed herein.
• An ASIC for retrieving an estimate 1114 may correspond to, for example, a processor and/or one or more other components as discussed herein.
• An ASIC for using a denoising process 1116 may correspond to, for example, a denoiser and/or one or more other components as discussed herein.
• these components may be implemented via appropriate processor components. These processor components may in some aspects be implemented, at least in part, using structure as taught herein. In some aspects a processor may be adapted to implement a portion or all of the functionality of one or more of these components. In some aspects one or more of the components, especially components represented by dashed boxes, are optional.
• apparatuses 800, 900, 1000, and 1100 may comprise one or more integrated circuits.
• a single integrated circuit may implement the functionality of one or more of the illustrated components, while in other aspects more than one integrated circuit may implement the functionality of one or more of the illustrated components.
• FIGS. 5A and 5B may be implemented using any suitable means. Such means also may be implemented, at least in part, using corresponding structure as taught herein.
• the components described above in conjunction with the "ASIC for" components of FIGS. 5 A and 5B also may correspond to similarly designated “means for” functionality.
• one or more of such means may be implemented using one or more of processor components, integrated circuits, or other suitable structure as taught herein.
• the methods described and claimed may be carried, at least in part, using software modules carried out by suitable processing means.
• any reference to an element herein using a designation such as "first,” “second,” and so forth does not generally limit the quantity or order of those elements. Rather, these designations may be used herein as a convenient method of distinguishing between two or more elements or instances of an element. Thus, a reference to first and second elements does not mean that only two elements may be employed there or that the first element must precede the second element in some manner. Also, unless stated otherwise a set of elements may comprise one or more elements.
• any of the various illustrative logical blocks, modules, processors, means, circuits, and algorithm steps described in connection with the aspects disclosed herein may be implemented as electronic hardware (e.g., a digital implementation, an analog implementation, or a combination of the two, which may be designed using source coding or some other technique), various forms of program or design code incorporating instructions (which may be referred to herein, for convenience, as "software” or a "software module”), or combinations of both.
• software or a “software module”
• the various illustrative logical blocks, modules, and circuits described in connection with the aspects disclosed herein may be implemented within or performed by an integrated circuit ("IC"), an access terminal, or an access point.
• the IC may comprise a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, electrical components, optical components, mechanical components, or any combination thereof designed to perform the functions described herein, and may execute codes or instructions that reside within the IC, outside of the IC, or both.
• a general purpose processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine.
• a processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.
• any specific order or hierarchy of steps in any disclosed process is an example of a sample approach. Based upon design preferences, it is understood that the specific order or hierarchy of steps in the processes may be rearranged while remaining within the scope of the present disclosure.
• the accompanying method claims present elements of the various steps in a sample order, and are not meant to be limited to the specific order or hierarchy presented.
• a software module e.g., including executable instructions and related data
• other data may reside in a data memory such as RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, a hard disk, a removable disk, a CD-ROM, or any other form of computer-readable storage medium known in the art.
• a sample storage medium may be coupled to a machine such as, for example, a computer/processor (which may be referred to herein, for convenience, as a "processor") such the processor can read information (e.g., code) from and write information to the storage medium.
• a sample storage medium may be integral to the processor.
• the processor and the storage medium may reside in an ASIC.
• the ASIC may reside in user equipment.
• the processor and the storage medium may reside as discrete components in user equipment.
• any suitable computer-program product may comprise a computer-readable medium comprising codes (e.g., executable by at least one computer) relating to one or more of the aspects of the disclosure.
• a computer program product may comprise packaging materials.

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